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A295960 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n) - 1, where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences.

%I A295960 #8 Feb 22 2018 22:42:51
%S A295960 1,3,8,16,30,54,93,157,261,430,704,1148,1868,3033,4919,7971,12910,
%T A295960 20902,33834,54759,88617,143401,232044,375472,607544,983046,1590621,
%U A295960 2573699,4164353,6738086,10902474,17640596,28543107,46183741,74726887,120910668,195637596
%N A295960 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n) - 1, where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences.
%C A295960 The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. a(n)/a(n-1) -> (1 + sqrt(5))/2 = golden ratio (A001622).
%C A295960 See A295862 for a guide to related sequences.
%H A295960 Clark Kimberling, <a href="/A295960/b295960.txt">Table of n, a(n) for n = 0..2000</a>
%H A295960 Clark Kimberling, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL10/Kimberling/kimberling26.html">Complementary equations</a>, J. Int. Seq. 19 (2007), 1-13.
%e A295960 a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, b(2) = 5
%e A295960 b(3) = 6 (least "new number")
%e A295960 a(2) = a(1) + a(0) + b(2) - 1 = 8
%e A295960 Complement: (b(n)) = (2, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, ...)
%t A295960 a[0] = 1; a[1] = 3; b[0] = 2; b[1] = 4; b[2] = 5;
%t A295960 a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n] - 1;
%t A295960 j = 1; While[j < 6, k = a[j] - j - 1;
%t A295960 While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t A295960 Table[a[n], {n, 0, k}];  (* A295960 *)
%Y A295960 Cf. A001622, A000045, A295862.
%K A295960 nonn,easy
%O A295960 0,2
%A A295960 _Clark Kimberling_, Dec 08 2017